# What are the details of this variational calculus solution?

Suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of bullet of a fixed area has the least drag?

This problem gives

$$\int {1\over 1+y'^2} + \lambda y dx$$

And the equation for y' you get is

$$y' = \lambda x (1 - 2 y'^2 - y'^4)$$

or

$$y = \int \lambda x ( 1 - 2 y^2 - y^4)$$

Which you can solve in a series by plugging in $y={\lambda x^2\over 2}$ and iterating a few times using the relation above as a recursion.

I am a beginner in calculus of variations and I can not understand the solution in that answer.

Is there any extra assumption in the problem's statement? What should the exact statement of the underlying physics/mathematics model be? I have difficulties in writing down the optimization equations of "has the least drag"

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 Hi craftsman.don - I've edited your question to make it more self-contained, and also to remove the second part of your question about simpler problems because (1) that's a list question, which isn't really appropriate for this site, and (2) it's best to keep each post about a single question. Please review the edit and fix it if the new version doesn't accurately represent what you wanted to ask. – David Zaslavsky♦ Oct 28 '12 at 12:11